A network reliability approach to the analysis of combinatorial repairable threshold schemes

Bailey Kacsmar; Douglas R. Stinson

doi:10.3934/amc.2019037

Article Contents

2019, Volume 13, Issue 4: 601-612. Doi: 10.3934/amc.2019037

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A network reliability approach to the analysis of combinatorial repairable threshold schemes

Bailey Kacsmar and
Douglas R. Stinson^,

David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

^* Corresponding author: Douglas R. Stinson
^* Corresponding author: Douglas R. Stinson

Received: October 31, 2018

Revised: December 31, 2018

Published: June 2019

The second author is supported by NSERC discovery grant RGPIN-03882

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Abstract

A repairable threshold scheme (which we abbreviate to RTS) is a $ (\tau,n) $-threshold scheme in which a subset of players can "repair" another player's share in the event that their share has been lost or corrupted. This will take place without the participation of the dealer who set up the scheme. The repairing protocol should not compromise the (unconditional) security of the threshold scheme. Combinatorial repairable threshold schemes (or combinatorial RTS) were recently introduced by Stinson and Wei [8]. In these schemes, "multiple shares" are distributed to each player, as defined by a suitable combinatorial design called the distribution design. In this paper, we study the reliability of these combinatorial repairable threshold schemes in a setting where players may not be available to take part in a repair of a given player's share. Using techniques from network reliability theory, we consider the probability of existence of an available repair set, as well as the expected number of available repair sets, for various types of distribution designs.

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Mathematics Subject Classification: Primary: 94A62; Secondary: 05B05, 90B25.

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Table 1. Expected number of repair sets for $ {\mathtt{SQS}}(v) $

size	type	expected number
2		$ 3(r_2-1)^2 p^2 $
3	pair-pair-pair	$ 4(r_2-1)^3 p^3 $
3	pair-pair-point	$ 12(r_2-1)^2(r_1-3r_2+2) p^3 $
3	pair-point-point	$ 6(r_2-1)(r_1-3r_2+2)^2 p^3 $
4		$ (r_1-3r_2+2)^4 p^4 $

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References

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